Explicit stationary distribution of the (L, 1)-reflecting random walk on the half line

Abstract

In this paper, we consider the (L, 1) state-dependent reflecting random walk (RW) on the half line, which is a RW allowing jumps to the left at a maxial size L. For this model, we provide an explicit criterion for (positive) recurrence and an explicit expression for the stationary distribution. As an application, we prove the geometric tail asymptotic behavior of the stationary distribution under certain conditions. The main tool employed in the paper is the intrinsic branching structure within the (L, 1)-random walk.

Tail asymptotic of the stationary distribution for the state dependent (1,R)-reflecting random walk: near critical

“…With the explicit expression of the stationary distribution (1.3) at hand, we can consider the tail asymptotic of the distribution. Firstly, it is not difficult (but is also not obviously, as [10] for the (L, 1)-RW) to see that the tail of π(i) is geometric decay in the sense lim i→∞ log π(i) i…”